Null Rotations over \HH'\otimes\CC

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Section A.4 Null Rotations over \(\HH'\otimes\CC\)

Consider \(3+1\)-dimensional Minkowski space, with coordinates \((T,X,Y,Z)\text{.}\) As usual, we can represent Minkowski 4-vectors as \(2\times2\) Hermitian matrices

\begin{equation} \XX_0 = \begin{pmatrix} T+Z & X-iY \\ X+iY & T-Z \\ \end{pmatrix} ,\tag{A.4.1} \end{equation}

or equivalently over \(\CC'\otimes\CC\) as

\begin{equation} \YY_0 = \begin{pmatrix} T\,L+Z & X-iY \\ X+iY & T\,L-Z \\ \end{pmatrix} ;\tag{A.4.2} \end{equation}

in either case, the norm is given by (\(-1\) times) the determinant, since

\begin{equation} \det\XX_0 = \det\YY_0 = T^2 - X^2 - Y^2 - Z^2 .\tag{A.4.3} \end{equation}

Extend this description to \(\so(4,2)\cong\su(2,\HH'\otimes\CC)\text{,}\) acting on matrices of the form

\begin{equation} \YY = \begin{pmatrix} Z+T\,L+p\,K+q\,KL & X-iY \\ X+iY & -Z+T\,L+p\,K+q\,KL \\ \end{pmatrix} ,\tag{A.4.4} \end{equation}

where now

\begin{equation} \det\YY = T^2 - X^2 - Y^2 - Z^2 - p^2 + q^2 .\tag{A.4.5} \end{equation}

Rotations in the \((X,P)\) plane, and similarly boosts in the \((X,Q)\) plane, are generated by

\begin{equation} r_X = \begin{pmatrix} 0 & K \\ K & 0 \\ \end{pmatrix} , \qquad b_X = \begin{pmatrix} 0 & KL \\ KL & 0 \\ \end{pmatrix} .\tag{A.4.6} \end{equation}

respectively. What happens if we add these elements of the Lie algebra \(\so(4,2)\) together? Exponentiating \(r_X\pm b_X\) is easy. Since

\begin{equation} (K\pm KL)^2 = K(1\pm L)K(1\pm L) = K(1\pm L)(1\mp L)K = 0 ,\tag{A.4.7} \end{equation}

we have

\begin{align} \exp{[(r_x\pm b_x)\alpha]} \amp= 1 + (r_x\pm b_x)\alpha\notag\\ \amp= \begin{pmatrix} 1 & (K\pm KL)\,\alpha \\ (K\pm KL)\,\alpha & 1 \\ \end{pmatrix} .\tag{A.4.8} \end{align}

Such transformations are called null rotations; they are neither rotations nor boosts. We can repeat this construction using any linear combination of \(X,Y,Z,T\) instead of \(X\text{;}\) the resulting null transformation is given by the group element

\begin{equation} 1 + (K\pm KL) \YY_0 \alpha = 1 \pm (K\pm KL) \XX_0 \alpha ,\tag{A.4.9} \end{equation}

with generator

\begin{equation} (K\pm KL) \YY_0 = \pm(K\pm KL) \XX_0 ,\tag{A.4.10} \end{equation}

where the equalities follow from the fact that \((K\pm KL)L=\pm(K\pm KL)\text{.}\)